1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "basic_pairs.ma".
16 include "o-basic_pairs.ma".
17 include "relations_to_o-algebra.ma".
19 (* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
20 definition o_basic_pair_of_basic_pair: basic_pair → Obasic_pair.
23 [ apply (map_objs2 ?? POW (concr b));
24 | apply (map_objs2 ?? POW (form b));
25 | apply (map_arrows2 ?? POW (concr b) (form b) (rel b)); ]
28 definition o_relation_pair_of_relation_pair:
29 ∀BP1,BP2. relation_pair BP1 BP2 →
30 Orelation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
33 [ apply (map_arrows2 ?? POW (concr BP1) (concr BP2) (r \sub \c));
34 | apply (map_arrows2 ?? POW (form BP1) (form BP2) (r \sub \f));
35 | apply (.= (respects_comp2 ?? POW (concr BP1) (concr BP2) (form BP2) r\sub\c (⊩\sub BP2) )^-1);
36 cut ( ⊩ \sub BP2 ∘ r \sub \c =_12 r\sub\f ∘ ⊩ \sub BP1) as H;
38 apply (respects_comp2 ?? POW (concr BP1) (form BP1) (form BP2) (⊩\sub BP1) r\sub\f);
42 lemma o_relation_pair_of_relation_pair_is_morphism :
43 ∀S,T:category2_of_category1 BP.
44 ∀a,b:arrows2 (category2_of_category1 BP) S T.a=b →
45 (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)))
46 (o_relation_pair_of_relation_pair S T a) (o_relation_pair_of_relation_pair S T b).
48 intros (a b Eab); split; unfold o_relation_pair_of_relation_pair; simplify;
49 unfold o_basic_pair_of_basic_pair; simplify;
50 [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
51 | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
52 | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
53 | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
56 apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (a\sub\c) (⊩\sub T))^-1);
58 apply (.= (respects_comp2 ?? POW (concr S) (concr T) (form T) (b\sub\c) (⊩\sub T))^-1);
64 lemma o_relation_pair_of_relation_pair_morphism :
65 ∀S,T:category2_of_category1 BP.
66 unary_morphism2 (arrows2 (category2_of_category1 BP) S T)
67 (arrows2 OBP (o_basic_pair_of_basic_pair S) (o_basic_pair_of_basic_pair T)).
70 [ apply (o_relation_pair_of_relation_pair S T);
71 | apply (o_relation_pair_of_relation_pair_is_morphism S T)]
74 lemma o_relation_pair_of_relation_pair_morphism_respects_id:
75 ∀o:category2_of_category1 BP.
76 o_relation_pair_of_relation_pair_morphism o o (id2 (category2_of_category1 BP) o)
77 = id2 OBP (o_basic_pair_of_basic_pair o).
78 simplify; intros; whd; split;
79 [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
80 | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
81 | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
82 | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
85 apply prop22;[2,4,6,8: apply rule #;]
86 apply (respects_id2 ?? POW (concr o));
89 lemma o_relation_pair_of_relation_pair_morphism_respects_comp:
90 ∀o1,o2,o3:category2_of_category1 BP.
91 ∀f1:arrows2 (category2_of_category1 BP) o1 o2.
92 ∀f2:arrows2 (category2_of_category1 BP) o2 o3.
93 (eq2 (arrows2 OBP (o_basic_pair_of_basic_pair o1) (o_basic_pair_of_basic_pair o3)))
94 (o_relation_pair_of_relation_pair_morphism o1 o3 (f2 ∘ f1))
96 (o_relation_pair_of_relation_pair_morphism o1 o2 f1)
97 (o_relation_pair_of_relation_pair_morphism o2 o3 f2)).
98 simplify; intros; whd; split;
99 [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
100 | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
101 | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
102 | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
105 apply prop22;[2,4,6,8: apply rule #;]
106 apply (respects_comp2 ?? POW (concr o1) (concr o2) (concr o3) f1\sub\c f2\sub\c);
109 definition BP_to_OBP: carr3 (arrows3 CAT2 (category2_of_category1 BP) OBP).
111 [ apply o_basic_pair_of_basic_pair;
112 | intros; apply o_relation_pair_of_relation_pair_morphism;
113 | apply o_relation_pair_of_relation_pair_morphism_respects_id;
114 | apply o_relation_pair_of_relation_pair_morphism_respects_comp;]
117 theorem BP_to_OBP_faithful:
118 ∀S,T.∀f,g:arrows2 (category2_of_category1 BP) S T.
119 map_arrows2 ?? BP_to_OBP ?? f = map_arrows2 ?? BP_to_OBP ?? g → f=g.
120 intros; change with ( (⊩) ∘ f \sub \c = (⊩) ∘ g \sub \c);
121 apply (POW_faithful);
122 apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) f \sub \c (⊩ \sub T));
124 apply (.= respects_comp2 ?? POW (concr S) (concr T) (form T) g \sub \c (⊩ \sub T));
129 theorem BP_to_OBP_full:
130 ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
132 cases (POW_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
133 cases (POW_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
135 constructor 1; [apply gc|apply gf]
136 apply (POW_faithful);
137 apply (let xxxx ≝POW in .= respects_comp2 ?? POW (concr S) (concr T) (form T) gc (rel T));
138 apply rule (.= Hgc‡#);
139 apply (.= Ocommute ?? f);
141 apply (let xxxx ≝POW in (respects_comp2 ?? POW (concr S) (form S) (form T) (rel S) gf)^-1)]
143 [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
144 | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
145 | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
146 | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
147 simplify; apply (†(Hgc‡#));